What Is CFA? How to Read Results and Assess the Measurement Model
What Is CFA
CFA, short for Confirmatory Factor Analysis, is a technique used to test a measurement model that was specified in advance. The model identifies which observed variables belong to which latent constructs. For example, SAT1, SAT2, and SAT3 may all measure satisfaction, while TR1, TR2, and TR3 measure trust.
When you run CFA, you enter the theoretical structure into the software first and then check whether the actual data fit that structure. The output usually includes overall model fit, standardized factor loading, Composite Reliability, AVE, and discriminant validity between constructs.
CFA is usually performed in CB-SEM, commonly with AMOS, lavaan in R, or similar SEM software. The two-step procedure, testing the measurement model first and the structural model second, is presented in (Anderson and Gerbing, 1988). CFA therefore does not replace EFA. EFA explores possible structures in the data, while CFA tests the structure you have already hypothesized.
In practical terms, CFA estimates the relationship between a latent construct and its observed variables. If the latent construct is η and the observed variable is y, the measurement model can be written in general form as:
y = λη + ε
Here, λ is the factor loading and ε is measurement error. You do not need to calculate this formula by hand. Your job is to specify the model correctly, import clean data, and read the output tables correctly.
Why CFA Matters in Quantitative Research
CFA answers a direct question: do the observed variables you are using actually measure the constructs in your research model. For example, you may propose that four observed variables measure service quality and three observed variables measure usage intention. CFA tests that assumption using your survey data.
CFA results usually support four decisions. First, you check whether the model fits the data at the overall level. Second, you examine whether each observed variable has a sufficiently large loading and practical meaning. Third, you assess the reliability and convergent validity of each latent construct using CR and AVE. Fourth, you check whether the latent constructs are distinct from one another.
CFA is particularly useful when a scale is adapted from earlier research, the expected number of factors is already known, and you want to show that the structure fits a new research context. If you do not yet know how the observed variables will group into factors, review the EFA procedure first. After EFA produces factor groups, CFA can be used with another sample or at a later stage to test the structure.
You also need to distinguish CFA from regression. Regression tests relationships between variables, while CFA tests how latent constructs are measured. A regression model may have attractive p-values, but if the scale has not been established, conclusions about relationships between constructs still lack a sound basis.
What CFA Values Are Acceptable
No single number determines whether a CFA model is acceptable. You need to read model fit, loading, CR, AVE, and discriminant validity together. The thresholds below are commonly used reference points, not automatic reasons to delete variables.
| Assessment group | Indicator and reference threshold | Source in the approved list |
|---|---|---|
| CB-SEM model fit | CFI and TLI at 0.90 or above, RMSEA and SRMR no higher than 0.08 | (Hu and Bentler, 1999) |
| Factor loading | 0.50 or above, with higher loading usually providing stronger evidence | (Hair et al., 2010) |
| Composite Reliability | CR at 0.70 or above | (Fornell and Larcker, 1981) |
| Average Variance Extracted | AVE at 0.50 or above | (Fornell and Larcker, 1981) |
| Discriminant validity | The square root of AVE is greater than the correlation with another construct | (Fornell and Larcker, 1981) |
| HTMT | Below 0.85, or below 0.90 for conceptually similar constructs | (Henseler et al., 2015) |
For CFA in CB-SEM, you may see indicators such as Chi-square, degrees of freedom, CMIN/DF, CFI, TLI, RMSEA, and SRMR. When reporting the results, do not show one acceptable indicator and then declare the entire model good. Present the main indicators, state the thresholds you selected, and explain any indicator that does not meet them.
Hair et al. use a factor loading of 0.50 or above as a reference point (Hair et al., 2010). CR from 0.70 and AVE from 0.50 are used to assess composite reliability and convergent validity (Fornell and Larcker, 1981). If the model is exploratory or the scale is new, explain the research context instead of applying one threshold mechanically.
The CFI, TLI, RMSEA, and SRMR thresholds in the table above apply to CB-SEM (Hu and Bentler, 1999). Do not use this group of indicators to assess a PLS-SEM model in SmartPLS. For PLS-SEM, the indicators you commonly report include outer loading, CR, AVE, HTMT, VIF, R², f², and Q².
How to Read CFA Output
The table names depend on the software you use. In AMOS, you will usually review Model Fit Summary, Regression Weights, Standardized Regression Weights, Squared Multiple Correlations, and covariance tables. In R, the table names depend on the analysis package, but the reading logic still centers on the same groups of indicators.
The following is illustrative output, not the result of a real study. The table is shortened in the format students commonly use after running CFA.
| Result group | Indicator or variable | Illustrative Estimate | Reference threshold | How to read it |
|---|---|---|---|---|
| Model Fit Summary | CMIN/DF | 2.41 | Must be considered with other fit indicators | The general discrepancy between the data and the model |
| Model Fit Summary | CFI | 0.934 | ≥ 0.90 | The model's incremental fit |
| Model Fit Summary | TLI | 0.921 | ≥ 0.90 | A fit indicator adjusted for complexity |
| Model Fit Summary | RMSEA | 0.061 | ≤ 0.08 | The model's approximate error |
| Model Fit Summary | SRMR | 0.052 | ≤ 0.08 | The standardized residual error |
| Standardized Regression Weights | SAT2 <--- SAT | 0.782 | ≥ 0.50 | SAT2 meets the reference loading threshold |
| Standardized Regression Weights | SAT3 <--- SAT | 0.846 | ≥ 0.50 | SAT3 has a fairly clear relationship with SAT |
| Reliability summary | SAT, CR | 0.887 | ≥ 0.70 | Composite reliability meets the reference threshold |
| Reliability summary | SAT, AVE | 0.663 | ≥ 0.50 | Convergent validity meets the reference threshold |
Read the output from the overall model to the individual indicators. Start with Model Fit Summary. In the example, CFI and TLI are both above 0.90, while RMSEA and SRMR are below 0.08. These are illustrative values placed beside the thresholds from (Hu and Bentler, 1999), not evidence that your model will produce the same values.
Next, open Standardized Regression Weights and read the standardized Estimate column. The arrow SAT2 <--- SAT indicates that SAT2 is explained by the latent construct SAT. A loading of 0.782 means that this variable has a substantial measurement relationship with SAT in the example. You also need to check the p-value or the statistical significance indicator shown by the software.
Then summarize CR and AVE for each latent construct. CR and AVE do not usually appear directly in AMOS's default tables, so you may need to calculate them from the standardized loadings or prepare a separate summary table. Do not confuse AVE with the percentage of variance extracted in EFA. AVE assesses the convergent validity of one latent construct in the measurement model.
Finally, check the correlations between latent constructs and HTMT if your procedure reports it. A model can have good overall fit and still have a discriminant validity problem if two constructs measure almost the same content.
What to Do When CFA Does Not Meet the Thresholds
Before deleting a variable, identify exactly which indicator is unacceptable. A low loading, high RMSEA, high HTMT, and a variable with a large error are different situations, so they require different responses.
Check the Data and the Entered Model
Reopen the data file and check variable names, the direction of reverse-coded variables, missing data, and Likert-scale coding. A reverse-coded variable entered in the wrong direction or a column read as text can distort the results. Check that the CFA model connects each observed variable to the latent construct specified in the original hypotheses.
Review Variables with Low Loadings
If a loading is low, read the wording of the observed variable and check whether it really belongs to the construct being measured. You can consider deleting the variable when the theory, measurement content, and statistical results support that decision. Repeatedly deleting variables until the fit indicators look good, without recording the reasons, is likely to create questions from the committee.
Check Modification Indices for a Theoretical Reason
Modification Indices may suggest adding a relationship between error terms or changing the model structure. Do not connect errors simply because the software reports a large number. Make an adjustment only when two variables have very similar content or there is a clear measurement reason. Then record the original model, the adjusted model, and the reason for the change.
Reconsider the Scale Structure
If several indicators fail at the same time, the problem may lie in the scale structure or the research context rather than in one variable. Recheck the theoretical foundation, translation, wording, and survey sample. You may need to run the adjusted measurement model again, but every change must be reported transparently.
EFA, CFA, and principal component analysis are not three names for the same procedure. PCA reduces data dimensions, while EFA and CFA are more directly concerned with exploring or testing factor structures. You can also review communality and eigenvalue when reading EFA output, but those indicators do not replace CFA assessment.
Distinguishing CFA from EFA and PCA
CFA starts with an existing model. You decide in advance which variables belong to which factor and then test how well the model fits. This is why CFA usually appears after the theoretical foundation and before testing hypothesized relationships.
EFA starts more openly. The software identifies groups of variables from the data, usually when the scale structure is unclear or when you want to examine a new context. EFA commonly includes KMO, Bartlett's Test, Total Variance Explained, and Rotated Component Matrix. If you are looking at the rotated matrix, you are most likely running EFA rather than CFA.
PCA finds composite components to reduce the number of dimensions, while CFA models latent constructs and measurement error. Therefore, do not use PCA results to claim that you confirmed the measurement model with CFA.
In a thesis, you can present the logic as follows: the theoretical foundation proposes the constructs and observed variables, EFA is used when structure needs to be explored during the preliminary stage, CFA tests the measurement model, and the structural model then tests hypothesized effects. The exact sequence depends on the research design and the software you use.
Common Errors
The first error is calling every factor analysis CFA. If you only run Analyze > Dimension Reduction > Factor in SPSS and read the Rotated Component Matrix, you are usually looking at EFA or PCA, not CFA in the CB-SEM sense.
The second error is using PLS-SEM thresholds for AMOS, or using CFI, TLI, and RMSEA to assess SmartPLS. Identify whether you are using CB-SEM or PLS-SEM before choosing the set of indicators to report.
The third error is looking only at the p-value of Chi-square. Chi-square can be sensitive to sample size, so you need to read it together with CFI, TLI, RMSEA, SRMR, and the model structure. Do not make the conclusion from one number alone.
The fourth error is deleting variables in batches based on Modification Indices. Each deleted variable changes the content of the construct. Save each version of the output and state the reason for deletion in the methods or results section.
The fifth error is reporting CR and AVE without explaining how they were calculated and without checking discriminant validity. A complete CFA table should show both overall model fit and the quality of each latent construct.
How to write this in your thesis: “The CFA results show that the measurement model has the following fit indicators: CFI = [value], TLI = [value], RMSEA = [value], and SRMR = [value]. The factor loadings range from [lowest value] to [highest value], CR for the constructs is at least [value], and AVE is at least [value], providing evidence of reliability and convergent validity according to the selected criteria.” Replace every placeholder with your actual output and cite the appropriate threshold source.
Frequently asked questions
What is CFA and when should you use it?
CFA stands for Confirmatory Factor Analysis and is used to test a measurement model specified in advance. You use CFA when you have a theoretical foundation or an established scale, know which observed variables belong to each latent construct, and want to test that structure with survey data.
Can you run CFA in SPSS?
Standard SPSS supports EFA and regression, but it is not the main tool for running CFA in CB-SEM. CFA is usually run with AMOS, R using an SEM package, or a comparable SEM program. You can still use SPSS to clean the data before transferring it to a CFA tool.
What CFA values are acceptable?
You need to assess a group of criteria, including model fit, factor loading, CR, AVE, and discriminant validity. Common reference points include CFI and TLI at 0.90 or above, RMSEA and SRMR no higher than 0.08 (Hu and Bentler, 1999), CR from 0.70, and AVE from 0.50 (Fornell and Larcker, 1981). These are reference points that must be interpreted in the context of the model, sample size, and theoretical foundation.
Do you need to collect more data when CFA does not meet the thresholds?
Not necessarily. First check data-entry errors, reverse-coded variables, missing data, the model structure, and the wording of the variables. If the model is unsuitable because the scale does not fit the context, collecting more data with the same questionnaire may not solve the problem. After a reasonable review, if the evidence is still insufficient, discuss whether to adjust the model or scale with your supervisor.
How is CFA different from EFA?
EFA explores a factor structure from the data, while CFA tests a structure specified in advance. EFA commonly uses KMO, Bartlett's Test, and the rotated matrix. CFA commonly uses Model Fit, standardized loading, CR, AVE, and discriminant validity. Both procedures may appear in the same thesis, but EFA results should not be called CFA results.
Open your data file again, identify whether you are running CB-SEM or PLS-SEM, and create a separate table containing Model Fit, loading, CR, AVE, and discriminant validity before writing the results chapter. If you need to run CFA on a .sav or .csv file and check each adjustment round, you can use M4 data analysis.